Question

设 $a$ 和 $b$ 是实数。多项式 $x^3 + x^2 + ax + b$ 除以 $x - 1$ 时余数为 $4$ ，除以 $x - 2$ 时余数为 $6$ 。 $b - a$ 等于多少？

Suppose $a$ and $b$ are real numbers. When the polynomial $x^3 + x^2 + ax + b$ is divided by $x - 1,$ the remainder is $4.$ When the polynomial is divided by $x - 2,$ the remainder is $6.$ What is $b - a?$

$14$

$15$

$16$

$17$

$18$

答案：E

知识点：多项式方程组

难度评级：1250

解答：

由余数定理，直接代入即可。我们有 $p(1) = 1 + 1 + a + b = 4$ ，所以 $a + b = 2$ 。又有 $p(2) = 8 + 4 + 2a + b = 6$ ，所以 $2a + b = -6$ 。第二式减去第一式，得 $a = -8$ ，从而 $b = 10$ 。因此 $b - a = 10 - (-8) = 18$ ，所以正确答案是 E。

By the Remainder Theorem, just plug in. We get $p(1) = 1 + 1 + a + b = 4,$ so $a + b = 2.$ And $p(2) = 8 + 4 + 2a + b = 6,$ so $2a + b = -6.$ Subtract the first from the second: $a = -8,$ hence $b = 10.$ Then $b - a = 10 - (-8) = 18.$ Thus, E is the correct answer.

2025 AMC 10A 第 7 题

答案：E

难度评级：1250

解答：

其他年份的第 7 题